Post
by The Orange One » 06 Jun 2016, 02:39
Actually (adapting the definition for rings), an integer p is prime if it satisfies the following:
1) p is not zero
2) p is not a unit - i.e. there is no integer q such that pq=1
3) If p divides ab, then p divides a or p divides b - where a and b are both integers.
A similar concept to primality is irreducibility. An integer p is irreducible if:
1) p is not zero
2) p is not a unit - i.e. there is no integer q such that pq=1
3) If p=ab, with a and b both integers, then either a or b is a unit.
So, under the definition:
• 0 is not prime (it is zero)
• 1 is not prime (it is a unit)
• 2 is prime
• -2 is prime
• 4 is not prime (4 divides 6 x 2 = 12, but does not divide either 6 or 2)
to name a few examples!
In the integers, primes and irreducibles are one and the same. In all rings, p is prime implies p is irreducible, but there are rings where p can be irreducible but not prime!
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